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S(m)=1+1/3 +...+1/m

Source: 2016 Argentina OMA Finals L3 p2

January 16, 2023
SumalgebraCompare

Problem Statement

For an integer m≥3m\ge 3, let S(m)=1+13+…+1mS(m)=1+\frac{1}{3}+…+\frac{1}{m} (the fraction 12\frac12 does not participate in addition and does participate in fractions 1k\frac{1}{k} for integers from 33 until mm). Let n≥3n\ge 3 and k≥3 k\ge 3 . Compare the numbers S(nk)S(nk) and S(n)+S(k)S(n)+S(k) .
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